The Exciting Universe Of Music Theory
presents

more than you ever wanted to know about...

Scale 3539: "Aeoryllic"

Scale 3539: Aeoryllic, Ian Ring Music Theory

Bracelet Diagram

The bracelet shows tones that are in this scale, starting from the top (12 o'clock), going clockwise in ascending semitones. The "i" icon marks imperfect tones that do not have a tone a fifth above. Dotted lines indicate axes of symmetry.

Tonnetz Diagram

Tonnetz diagrams are popular in Neo-Riemannian theory. Notes are arranged in a lattice where perfect 5th intervals are from left to right, major third are northeast, and major 6th intervals are northwest. Other directions are inverse of their opposite. This diagram helps to visualize common triads (they're triangles) and circle-of-fifth relationships (horizontal lines).

Common Names

Zeitler
Aeoryllic
Dozenal
Weyian

Analysis

Cardinality

Cardinality is the count of how many pitches are in the scale.

8 (octatonic)

Pitch Class Set

The tones in this scale, expressed as numbers from 0 to 11

{0,1,4,6,7,8,10,11}

Forte Number

A code assigned by theorist Allen Forte, for this pitch class set and all of its transpositional (rotation) and inversional (reflection) transformations.

8-Z29

Rotational Symmetry

Some scales have rotational symmetry, sometimes known as "limited transposition". If there are any rotational symmetries, these are the intervals of periodicity.

none

Reflection Axes

If a scale has an axis of reflective symmetry, then it can transform into itself by inversion. It also implies that the scale has Ridge Tones. Notably an axis of reflection can occur directly on a tone or half way between two tones.

none

Palindromicity

A palindromic scale has the same pattern of intervals both ascending and descending.

no

Chirality

A chiral scale can not be transformed into its inverse by rotation. If a scale is chiral, then it has an enantiomorph.

yes
enantiomorph: 2423

Hemitonia

A hemitone is two tones separated by a semitone interval. Hemitonia describes how many such hemitones exist.

5 (multihemitonic)

Cohemitonia

A cohemitone is an instance of two adjacent hemitones. Cohemitonia describes how many such cohemitones exist.

3 (tricohemitonic)

Imperfections

An imperfection is a tone which does not have a perfect fifth above it in the scale. This value is the quantity of imperfections in this scale.

3

Modes

Modes are the rotational transformations of this scale. This number does not include the scale itself, so the number is usually one less than its cardinality; unless there are rotational symmetries then there are even fewer modes.

7

Prime Form

Describes if this scale is in prime form, using the Rahn/Ring formula.

no
prime: 751

Generator

Indicates if the scale can be constructed using a generator, and an origin.

none

Deep Scale

A deep scale is one where the interval vector has 6 different digits.

no

Interval Structure

Defines the scale as the sequence of intervals between one tone and the next.

[1, 3, 2, 1, 1, 2, 1, 1]

Interval Vector

Describes the intervallic content of the scale, read from left to right as the number of occurences of each interval size from semitone, up to six semitones.

<5, 5, 5, 5, 5, 3>

Interval Spectrum

The same as the Interval Vector, but expressed in a syntax used by Howard Hanson.

p5m5n5s5d5t3

Distribution Spectra

Describes the specific interval sizes that exist for each generic interval size. Each generic <g> has a spectrum {n,...}. The Spectrum Width is the difference between the highest and lowest values in each spectrum.

<1> = {1,2,3}
<2> = {2,3,4,5}
<3> = {3,4,5,6}
<4> = {5,6,7}
<5> = {6,7,8,9}
<6> = {7,8,9,10}
<7> = {9,10,11}

Spectra Variation

Determined by the Distribution Spectra; this is the sum of all spectrum widths divided by the scale cardinality.

2.25

Maximally Even

A scale is maximally even if the tones are optimally spaced apart from each other.

no

Maximal Area Set

A scale is a maximal area set if a polygon described by vertices dodecimetrically placed around a circle produces the maximal interior area for scales of the same cardinality. All maximally even sets have maximal area, but not all maximal area sets are maximally even.

no

Interior Area

Area of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle, ie a circle with radius of 1.

2.616

Polygon Perimeter

Perimeter of the polygon described by vertices placed for each tone of the scale dodecimetrically around a unit circle.

6.002

Myhill Property

A scale has Myhill Property if the Interval Spectra has exactly two specific intervals for every generic interval.

no

Balanced

A scale is balanced if the distribution of its tones would satisfy the "centrifuge problem", ie are placed such that it would balance on its centre point.

no

Ridge Tones

Ridge Tones are those that appear in all transpositions of a scale upon the members of that scale. Ridge Tones correspond directly with axes of reflective symmetry.

none

Propriety

Also known as Rothenberg Propriety, named after its inventor. Propriety describes whether every specific interval is uniquely mapped to a generic interval. A scale is either "Proper", "Strictly Proper", or "Improper".

Improper

Heteromorphic Profile

Defined by Norman Carey (2002), the heteromorphic profile is an ordered triple of (c, a, d) where c is the number of contradictions, a is the number of ambiguities, and d is the number of differences. When c is zero, the scale is Proper. When a is also zero, the scale is Strictly Proper.

(30, 60, 141)

Common Triads

These are the common triads (major, minor, augmented and diminished) that you can create from members of this scale.

* Pitches are shown with C as the root

Triad TypeTriad*Pitch ClassesDegreeEccentricityCloseness Centrality
Major TriadsC{0,4,7}341.9
E{4,8,11}242.1
F♯{6,10,1}242.3
Minor Triadsc♯m{1,4,8}341.9
em{4,7,11}341.9
Augmented TriadsC+{0,4,8}341.9
Diminished Triadsc♯°{1,4,7}242.1
{4,7,10}242.1
{7,10,1}242.3
a♯°{10,1,4}242.1
Parsimonious Voice Leading Between Common Triads of Scale 3539. Created by Ian Ring ©2019 C C C+ C+ C->C+ c#° c#° C->c#° em em C->em c#m c#m C+->c#m E E C+->E c#°->c#m a#° a#° c#m->a#° e°->em e°->g° em->E F# F# F#->g° F#->a#°

view full size

Above is a graph showing opportunities for parsimonious voice leading between triads*. Each line connects two triads that have two common tones, while the third tone changes by one generic scale step.

Diameter4
Radius4
Self-Centeredyes

Modes

Modes are the rotational transformation of this scale. Scale 3539 can be rotated to make 7 other scales. The 1st mode is itself.

2nd mode:
Scale 3817
Scale 3817: Zoryllic, Ian Ring Music TheoryZoryllic
3rd mode:
Scale 989
Scale 989: Phrolyllic, Ian Ring Music TheoryPhrolyllic
4th mode:
Scale 1271
Scale 1271: Kolyllic, Ian Ring Music TheoryKolyllic
5th mode:
Scale 2683
Scale 2683: Thodyllic, Ian Ring Music TheoryThodyllic
6th mode:
Scale 3389
Scale 3389: Socryllic, Ian Ring Music TheorySocryllic
7th mode:
Scale 1871
Scale 1871: Aeolyllic, Ian Ring Music TheoryAeolyllic
8th mode:
Scale 2983
Scale 2983: Zythyllic, Ian Ring Music TheoryZythyllic

Prime

The prime form of this scale is Scale 751

Scale 751Scale 751: Epoian, Ian Ring Music TheoryEpoian

Complement

The octatonic modal family [3539, 3817, 989, 1271, 2683, 3389, 1871, 2983] (Forte: 8-Z29) is the complement of the tetratonic modal family [139, 353, 1553, 2117] (Forte: 4-Z29)

Inverse

The inverse of a scale is a reflection using the root as its axis. The inverse of 3539 is 2423

Scale 2423Scale 2423: Otuian, Ian Ring Music TheoryOtuian

Enantiomorph

Only scales that are chiral will have an enantiomorph. Scale 3539 is chiral, and its enantiomorph is scale 2423

Scale 2423Scale 2423: Otuian, Ian Ring Music TheoryOtuian

Transformations:

In the abbreviation, the subscript number after "T" is the number of semitones of tranposition, "M" means the pitch class is multiplied by 5, and "I" means the result is inverted. Operation is an identical way to express the same thing; the syntax is <a,b> where each tone of the set x is transformed by the equation y = ax + b

Abbrev Operation Result Abbrev Operation Result
T0 <1,0> 3539       T0I <11,0> 2423
T1 <1,1> 2983      T1I <11,1> 751
T2 <1,2> 1871      T2I <11,2> 1502
T3 <1,3> 3742      T3I <11,3> 3004
T4 <1,4> 3389      T4I <11,4> 1913
T5 <1,5> 2683      T5I <11,5> 3826
T6 <1,6> 1271      T6I <11,6> 3557
T7 <1,7> 2542      T7I <11,7> 3019
T8 <1,8> 989      T8I <11,8> 1943
T9 <1,9> 1978      T9I <11,9> 3886
T10 <1,10> 3956      T10I <11,10> 3677
T11 <1,11> 3817      T11I <11,11> 3259
Abbrev Operation Result Abbrev Operation Result
T0M <5,0> 2549      T0MI <7,0> 1523
T1M <5,1> 1003      T1MI <7,1> 3046
T2M <5,2> 2006      T2MI <7,2> 1997
T3M <5,3> 4012      T3MI <7,3> 3994
T4M <5,4> 3929      T4MI <7,4> 3893
T5M <5,5> 3763      T5MI <7,5> 3691
T6M <5,6> 3431      T6MI <7,6> 3287
T7M <5,7> 2767      T7MI <7,7> 2479
T8M <5,8> 1439      T8MI <7,8> 863
T9M <5,9> 2878      T9MI <7,9> 1726
T10M <5,10> 1661      T10MI <7,10> 3452
T11M <5,11> 3322      T11MI <7,11> 2809

The transformations that map this set to itself are: T0

Nearby Scales:

These are other scales that are similar to this one, created by adding a tone, removing a tone, or moving one note up or down a semitone.

Scale 3537Scale 3537: Katogian, Ian Ring Music TheoryKatogian
Scale 3541Scale 3541: Racryllic, Ian Ring Music TheoryRacryllic
Scale 3543Scale 3543: Aeolonygic, Ian Ring Music TheoryAeolonygic
Scale 3547Scale 3547: Sadygic, Ian Ring Music TheorySadygic
Scale 3523Scale 3523, Ian Ring Music Theory
Scale 3531Scale 3531: Neveseri, Ian Ring Music TheoryNeveseri
Scale 3555Scale 3555: Pylyllic, Ian Ring Music TheoryPylyllic
Scale 3571Scale 3571: Dyrygic, Ian Ring Music TheoryDyrygic
Scale 3475Scale 3475: Kylian, Ian Ring Music TheoryKylian
Scale 3507Scale 3507: Maqam Hijaz, Ian Ring Music TheoryMaqam Hijaz
Scale 3411Scale 3411: Enigmatic, Ian Ring Music TheoryEnigmatic
Scale 3283Scale 3283: Mela Visvambhari, Ian Ring Music TheoryMela Visvambhari
Scale 3795Scale 3795: Epothyllic, Ian Ring Music TheoryEpothyllic
Scale 4051Scale 4051: Ionilygic, Ian Ring Music TheoryIonilygic
Scale 2515Scale 2515: Chromatic Hypolydian, Ian Ring Music TheoryChromatic Hypolydian
Scale 3027Scale 3027: Rythyllic, Ian Ring Music TheoryRythyllic
Scale 1491Scale 1491: Namanarayani, Ian Ring Music TheoryNamanarayani

This scale analysis was created by Ian Ring, Canadian Composer of works for Piano, and total music theory nerd. Scale notation generated by VexFlow, graph visualization by Graphviz, and MIDI playback by MIDI.js. All other diagrams and visualizations are © Ian Ring. Some scale names used on this and other pages are ©2005 William Zeitler (http://allthescales.org) used with permission.

Pitch spelling algorithm employed here is adapted from a method by Uzay Bora, Baris Tekin Tezel, and Alper Vahaplar. (An algorithm for spelling the pitches of any musical scale) Contact authors Patent owner: Dokuz Eylül University, Used with Permission. Contact TTO

Tons of background resources contributed to the production of this summary; for a list of these peruse this Bibliography. Special thanks to Richard Repp for helping with technical accuracy, and George Howlett for assistance with the Carnatic ragas.